Thin elastic sheets, such as those made of paper, crumple by developing kinks, creases, folds, and singular stress fields in response to constrained deformations, e.g., while squeezing them inside a hollow sphere or crushing them in our fist [1]}, [2]}, [3]}, [4]}, [5]}. The singularities persist even after the constraints are released indicating that the incurred deformations are plastic in nature. Similarly, an isolated defect, such as a disclination or a dislocation, in an elastic sheet (e.g., two-dimensional (2D) crystalline and amorphous materials) yields a deformed shape with a cone like singularity and a singular stress both at the defect location [6]}, [7]}, [8]}, [9]}. In this case the singularity appears due to an internal source of strain incompatibility (in the form of a defect) even in the absence of any external constraints and forces. The singular deformations and stress fields can also appear in response to concentrations of external forces and force couples, and singular sources of metric anomalies (e.g., thermal and growth strains).
In the present work, we model elastic sheets as von Kármán plates and use the theory of distributions to develop a general theoretical framework within which we can discuss a large variety of problems where singularities emerge at isolated points and curves in the plate domain. The existing von Kármán equations, which assume the fields to be smooth, are extended by including equations which should be satisfied at singular points and singular curves. The sources of inhomogeneity, given in terms of plastic strains, defect densities, and body force fields, are accordingly allowed to be singular. We also allow for several fields to develop concentrations over the singular subdomain. To the best of our knowledge, such a generalization of the von Kármán framework has not been attempted previously in the literature.
